Birth Control and Female Empowerment:
An Equilibrium Analysis1
Pierre-André Chiappori2
Sonia Oreffice3
April 2007
Please do not quote
1
Paper presented at seminars at Boston University, CEMFI, University of
Chicago, Columbia University, Hunter College, Clemson University, University
of British Columbia, University of Victoria, and at the 2005 World Congress of
the Econometric Society. We thank the participants and Gary Becker, Guillermo
Caruana, Kevin Lang, Claudio Michelacci, Javier Suarez and Derek Neal for many
helpful discussions. Financial support from the NSF (award # SES-05-32398) is
gratefully acknowledged. Errors are ours.
2
E. Rowan and Barbara Steinschneider Professor of Economics, Columbia University. Email: pc2167@columbia.edu
3
Department of Economics, Clemson University. Email: oreffic@clemson.edu
Abstract
It has been widely argued that innovations in birth control technology during the last decades have affected not only women’s fertility choices, but
generally their position in families and society. We analyze, from a theoretical perspective, the impact of these innovations on the intrahousehold
allocation of resources. We consider a model of frictionless matching on the
marriage market in which men, as well as women, differ in their preferences
for children; moreover, women may choose to have a child of their own,
whereas men must marry to enjoy fatherhood. We show that an improvement in the birth control technology generally increases the ’power’, hence
the welfare, of all women, including those who are not interested in the
new technology. However, this ’empowerment’ effect crucially relies on the
availability of the new technology to single women. Should the innovation
be reserved to married women (as was initially the case for the pill), the
conclusion may be reversed, resulting in a ’disempowerment’ effect. Various
extensions are discussed.
1
Introduction
The innovations in birth control during the last four decades have had a
strong impact on modern societies. The direct effect on demography, and
especially on birth rates in specific social groups, has been abundantly studied.1 An additional, indirect effect operates through the reduction in uncertainty faced by couples and single women due to the increased ability to plan
demographic phenomena. This fact has dramatically altered the context in
which decisions regarding human capital accumulation were made. For instance, many authors have argued that the spectacular increase in women’s
education and participation to the labor market since the 60s was in part
the consequence of these innovations.2
A third, and relatively less studied consequence goes through a different
channel - namely, the impact of the changes on the ‘balance of power’ within
the couple. After all, innovations in birth control technology, including the
legalization of abortion, have a potentially huge effect on men’s and women’s
respective decision rights within the household regarding such crucial issues
as the number or timing of births. It is hard to believe that a shift of such
magnitude would leave the balance of powers unaltered. This claim has
1
See for instance Levine et al. (1999), Klerman (1999), Angrist and Evans (1996) and
Oltmans Ananat, Gruber and Levine (2004).
2
See Michael (2000) and Goldin and Katz (2002).
1
been put forth by a number of sociologists3 , but there is little in terms of
an economic analysis of this phenomenon.
Oreffice (2007) has provided an empirical study of the ‘empowerment’
issue based on the collective approach to household behavior. The basic idea
is to check whether the legalization of abortion has affected the labor supply
of married females in their fertile age. If, as is often argued, legalization actually improves the wife’s bargaining situation, then she should, everything
equal, receive a larger share of household resources. By a standard income
effect, the reform should thus decrease her labor supply and increase her
husband’s. As it turns out, the predictions of the model are well confirmed
by the data. A significant effect is observed on both male and female labor
supply for married couples when the wife is in her fertile age, but not for
older couples nor for single males, while the effect for (young) single female
is small.
Although the claim that abortion legalization influenced intrahousehold
bargaining has received some empirical support, the mechanism by which
this empowerment occurs still deserves some scrutiny. While it is not hard
to convince oneself that some women will gain, whether all will is another
matter. A strong objection is that women have heterogeneous preferences
3
See for instance Héritier (2002).
2
for fertility (or different attitudes toward abortion); some, in particular, do
not consider abortion as an option, either for religious and ethical reasons
or because they do want children. Whether the legalization will benefit
these women as well is not clear. From a general perspective, the new
context will affect the matching process on the market for marriage, and
in particular the way the surplus generated by marriage is shared between
members. In principle, such ‘general equilibrium’ effects could annihilate or
even reverse the direct impact of the reform, particularly for these women
who are unlikely to derive much direct benefit from it.
The goal of the present paper is precisely to construct an explicit model
of matching on the marriage market in which these phenomena can be formally analyzed. We use a barebone setting in which any woman who wants
a child is biologically able to have one, but unwanted pregnancies are possible.4 Children (whether wanted or not) are a public good, in the general
sense that they affect both parent’s utility, although the father derives utility
from children only if married to their mother. This impact, however, is not
uniform; both male and female populations are heterogeneous in their preferences towards children, and some individuals may even dislike the presence
4
Our simple, static framework cannot address such dynamic issues as the timing of
birth. In particular, we only model the decision to have a child (or not). Depending on
the context, this can be interpreted as postponing the time of birth.
3
of children. In addition, motherhood decreases a woman’s ability to earn
income, which may discourage some women from becoming mothers unless
they receive an adequate compensation. Marriage is modelled as the outcome of a frictionless matching process; utilities are assumed transferable,
so that the impact of the matching process on individual utilities and intrahousehold transfers can readily be assessed. Finally, households are able to
commit, at least partially, on the division of resources during marriage. It
follows that the socioeconomic environment at the date of marriage, which
determines the agreement reached between spouses when they marry, has
potentially lasting effects on intrahousehold allocation. In particular, fertility decisions and the allocation of resources within the couple are closely
related issues; in our setting, they are jointly determined at the equilibrium
of the matching game.
A crucial ingredient of our approach is the legal and biological asymmetry between men and women regarding children. Namely, women do not
need to be married to become a mother, while men cannot enjoy paternity
without being in a couple.5 Therefore, those men for whom children are a
5
The anthropological literature has abundantly emphasized that an important feature
of marriage is the transfer of custodial rights on children to the father; see for instance
Bohannan 1949 (who distinguishes between between uxorial and genetricial rights), Bohannan and Middleton (1968), and Grossbard (1976). Similarly, the legal definition of
marriage typically emphasizes the link it establishes between the husband and the children
borne by the wife (see Posner 1994). For an economic analysis, see Grossbard-Schechtman
4
public good are potentially willing to compensate their wife for childbearing.
Whether such a compensation actually takes place at equilibrium depends
on the situation on the marriage market. If ‘marriageable’ men (defined, in
our setting, as men with a positive taste for fatherhood) are in very short
supply, in the sense that all married women would be willing to have a
child even if they were single (a situation we call ‘father shortage’), then
the surplus generated by marriage is entirely appropriated by the husband.
In the alternative situation, some married women would not elect to have
a child if single; they must therefore be compensated for childbearing, via
appropriate intrahousehold transfers. But then equilibrium conditions require that all women receive a share of the surplus generated in their couple.
Both the nature of the equilibrium and the division of the surplus are generated endogenously at equilibrium, and respond to changes in the economic,
demographic and technological environment.
In this framework, we consider the introduction of a new technology
that allows women (or couples) to better control fertility at no cost. Note
that our general setting can be applied not only to abortion, but to any
technological innovation affecting birth control. A particularly interesting
(1993) and especially Edlund and Korn, who explicitly analyze marriage as a ‘contract on
children in which custodial rights are transferred from the mother to the father’ (2002, p.
185).
5
example is provided by the pill, which first became available to married
women only, and later to singles as well. Our main finding is that the
legalization of abortion (and generally any improvement in birth control
that is freely available to all women) never worsens women’s well being
(and never increase men’s). In general, all women are strictly better off
after the reform (the exception being a situation of father shortage before or
after the reform). Even women who do not actually use the new technology
benefit from its introduction, because at equilibrium they receive a larger
share of household resources; in some cases, these women are actually those
who gain the most. Correspondingly, men are strict losers from the reform.
These conclusions remain essentially valid when individuals’ access to the
technology has a cost.
Our model thus predicts that a better birth control technology leads to an
empowerment of all women in the economy. However, this ‘empowerment
effect’ relies on a market adjustment mechanism in which single women
play a key role. Indeed, the reform benefits married women with children
only insofar as the new technology is available to single women as well.
Paradoxically, should the new technology be exclusively reserved to married
women (as was initially the case for the pill), then for most (married) women
the impact of the reform in terms of intrahousehold allocation is either nil
6
or even negative.
Finally, our framework generates interesting comparative statics results.
A proportional increase in female income (both with and without children)
decreases fertility; moreover, the gain for the wife typically exceed the additional income because of the intrahousehold redistribution triggered by the
change. A decline in the supply of marriageable men decreases total fertility but tends to increase out-of-wedlock births; it typically harms women’s
well-being. The introduction of more generous benefits targeted to single
mothers also increases out-of-wedlock fertility, but the impact on female
welfare is more complex, since its consequences on the qualitative patterns
of the stable match (and in particular on intrahousehold allocation of the
surplus generated by marriage) must be taken into account. We argue that
these features are broadly compatible with the main changes that took place
in the US over the last decades.
Related literature Our work is related to two lines of research. From a
theoretical perspective, modeling marriage markets as matching equilibria
is a standard approach, used in several contributions.6 Related, although
different, approaches rely on search models, in which initial matching is
6
Following Becker’s (1973, 1974, 1993) seminal work, one can mention, among others,
Mortensen (1988), Roth and Sotomayor (1990), Iyigun and Walsh (2004), Choo and Siow
(2005), and Chiappori, Iyigun and Weiss (2006).
7
generally random, but agents may decide to accept the current match or
continue searching;7 matching equilibria can be seen as limit cases of search
models when frictions become negligible (so that repeated sampling is costless). In both frameworks, the environment at the date of marriage is a
driving factor in the determination of intrahousehold allocation; as such,
they both assume that some minimum level of commitment is achievable
within couples.8
Our work is also related to the contribution of Akerlof, Yellen and Katz
(1996, from now on AYK) on out-of-wedlock child bearing in the United
States. The relationship between their approach and ours is discussed in
more detail at the end of this paper. At this stage, we may just emphasize
two crucial differences. First, while we allow for the existence of unwanted
pregnancies, we assume that at least a fraction of births are the outcome of
a deliberate decision made by a couple whose members derive utility from
children; in AYK, on the contrary, no woman (and no man) wants children,
even with marriage, and fertility is essentially an unwanted consequence of
sex. Secondly, the level of welfare reached by individuals within each couple
is taken as exogenously given in AYK, whereas it is fully endogenous in our
7
Examples include Burdett and Coles (1997, 1999), Aiyagari, Greenwood and Guner
(2000), Shimer and Smith (2000) and Chiappori and Weiss (2003, 2005).
8
An alternative view is based on intrahousehold bargaining in a no-commitment context; see for instance Lundberg and Pollak (1996) for a survey.
8
context. Our goal is precisely to analyze how equilibrium conditions on the
marriage market influence intrahousehold allocation of welfare.
Our approach is closer to that proposed by Neal (2004). In Neal’s model,
as in ours, marriage is modeled within a matching equilibrium framework.
Neal discusses how the conjunction of specific social policies (especially single mother benefits) and the situation on the labor market (particularly the
‘quality’ of the pool of potential husbands) may explain the raise in out-ofwedlock childbearing among less educated minority women during the last
decades. Neal’s framework, however, is quite different from ours, reflecting the difference in purpose between the two approaches. In Neal’s model,
women have identical preferences; the only source of heterogeneity between
males is in income. Our emphasis, unlike Neal’s, is on intrahousehold allocation, which, in a quasi-linear world, is not affected by male (or female)
income; income heterogeneity is thus irrelevant from our perspective. While
Neal concentrates on income heterogeneity, differences in taste for children
(and their consequences on fertility choices and intrahousehold allocations)
are the driving force in our framework. These aspects are crucial to explain
how birth control innovations affect marriage and fertility patterns. Finally,
Neal does not discuss the impact of abortion; in his model, in particular,
all married women elect to have children. Our approach can therefore be
9
viewed as complementary to Neal’s work.
The paper is organized as follows: Section 2 describes the model; Section
3 provides the equilibrium analysis and some comparative statics; Section 4
studies the effects of the technological innovation on the equilibrium outputs,
presents some extensions and discusses the links with the existing literature;
Section 5 concludes the paper.
2
The model
Preferences and budget constraints In the economy we consider, there
exists a continuum of men and women who derive utility from one private
composite good a and from children. For the sake of expositional simplicity,
we only consider the choice between having children or not, although the
generalization to different numbers of children is straightforward. Let the
dummy variable k denote the presence (k = 1) or the absence (k = 0) of children in the household; alternatively, k can be interpreted as the additional
child that men and women may have at any given moment.
Men have quasi-linear preferences over consumption and children. To
sharpen our analysis, we assume that the utility of single men only depends on their consumption; i.e., men cannot derive utility from (and do
not share the costs of) out-of-wedlock children, due to the fact that they
10
do not live in the same household.9 On the other hand, married men differ
in their preferences toward children. Specifically, their utility is of the form
UM (aM , k) = aM + v.k, where aM denotes male consumption of the private good; the individual-specific preference parameter v is distributed over
some support included in the interval [−V, V ], according to the atomless
distribution g. Note, in particular, that the utility of a married man does
not depend on the identity of the person he marries, but only on the joint
fertility decision and on the share of composite good he receives.10 Some
males may derive a negative utility from the presence of children; we shall
assume, however, that v > 0 for a positive mass of males. The total number
of males is M =
G (t) =
Rt
−V
RV
−V
g (t) dt, and we define the cumulative function G by
g (t) dt, so that G (V ) = M .
Similarly, female utility functions take the quasi-linear form U (aW , k) =
aW + u.k. Each woman is characterized by a preference parameter u, which
is distributed over some support included in some interval [−U, U ] according to the atomless density f ; let F denote the corresponding cumulative
distribution function, and let the total mass of women be normalized to 1.11
9
Our assumptions could be modified to allow for single men sharing some costs of
and deriving some utility from children who are not living with them; the qualitative
conclusions remain unchanged.
10
Following Chiappori and Weiss (2002), one could include an additional, match-specific
non-monetary benefit θi received by both spouses from the companionship in marriage.
11
The support of the distribution may be strictly included within the interval [−U, U ].
11
Note, in particular, that utility is transferable in this model.
We assume that taste parameters are publicly observable; when dating a
woman, a man acquires a reasonably good understanding of her attitude toward children (and conversely). Any woman (single or married) who wants
a child can have one. However, should she plan not to have children, unwanted births may occur with some probability p, which depends on the
technology available.12
Throughout the paper, the price of the private composite good is normalized to 1. Male income is denoted Y . Women initially receive an income
y; however, if a woman has children, her income drops to z, with z < y,
reflecting both the loss in her earning capacity due to childbearing and the
cost of raising the child. Hence a single woman without children consumes
her income y; if she decides to have a child (or if an unwanted pregnancy
occurs), she also consumes her income (which has dropped to z) and receives
moreover the utility u from her child.
It may be the case, for instance, that all women derive positive utility from children.
In general, we do not make assumptions on the respective values of males’ and females’
preferences for children (i.e., technically, on the distributions of u and v). Our model allows
for all women caring more for children than any man, or any alternative assumption.
12
Introducing uncertainty about the ability of having children would not change our
qualitative results. Indeed, women in general do not know whether they are biologically
able to have children until they try to have one; hence, there would be no hidden information or selection into marriage in this respect, and in our risk neutral world all payoffs can
always be interpreted as expected values. Also, our results are robust to the introduction
of different unwanted birth probabilities between married and single women.
12
In our framework, the couple’s decision to have a child or not will depend on both spouses’ preferences. Therefore the spouses must agree on
two issues. One is the fertility decision; i.e., they must decide whether to
have kids or not, a decision driven by their respective preferences towards
children. The other decision relates to the distribution of resources within
the household (i.e., the allocation of total income between male and female
consumption of the composite good).13 Both decision will be ultimately
determined by the equilibrium on the market for marriage.
As indicated above, our main focus is the impact of innovations in the
birth control technology. We model the legalization of abortion (and generally the availability of some birth control technology) as an exogenous
decrease in the probability p of experiencing an unwanted pregnancy.14
Marriage market We assume that the marriage market is frictionless;
i.e., we model marriage as a matching process, for which we will characterize the ’stable’ matches, defined by the property that no married person
would rather be single and there cannot be a man and a woman who would
13
Note that should the couple decide not to have children, each member’s consumption
may also be state-contingent, i.e. depend on whether a child is (unvoluntarily) born or
not. In our world, individuals are risk-neutral, and the payoffs can always be interpreted
as expected values.
14
In particular, we disregard here the other benefits derived from the improved birth
control technology (e.g., increased ability to plan and control human capital accumulation,
or easier access to sexual activity before marriage).
13
both prefer being married together than their current situation. In particular, any individual gets married as long as the utility (s)he can get from
marriage is larger than or equal to the utility (s)he gets from remaining
unmarried. Note that our analysis is consistent with a collective model of
household behavior in which spouses interact and take Pareto-efficient decisions, possibly transferring resources to each other. From this perspective,
a contribution of this paper, following several others15 , is to endogeneize the
’sharing rule’ that characterizes intrahousehold allocation.
3
Equilibrium on the marriage market
We now characterize the main features of the equilibrium (or stable assignment) reached on the marriage market.
3.1
Fertility and marriage decisions
Fertility We first consider fertility decisions, starting with single individuals. Single men consume their income, and get a utility equal to Y . Single
women, on the other hand, will decide to have children if and only if the
benefit compensates the income loss, i.e. if u ≥ y − z, leading to a util15
See for instance Browning, Chiappori and Weiss (2005), Iyigun and Walsh (2004), and
Chiappori, Iyigun and Weiss (2005).
14
ity equal to z + u. In the alternative case when u < y − z, single women
of type u choose not to have a child; any pregnancy will be involuntary
and occur with probability p. Their expected utility will thus be equal to
(1 − p) y + p (z + u). In what follows, the threshold y − z is denoted ū.
We call women whose parameter is larger than or equal to ū ‘unconditional
mothers’, since they want a child irrespective of their marital status. Other
women, however, will choose to have a child only if they are married and
their husband is willing to bear some of the costs; these women, whom we
call ‘conditional mothers’, are characterized by u < ū. We assume in what
follows that 0 < F (ū) < 1 - i.e., there exists both conditional and unconditional mothers in the population.
Regarding couples, note, first, that in a transferable utility context the
stable match must maximize total surplus. The total benefit to a couple
of having a child is v + u, whereas the cost is ū = y − z. It follows that
a married couple will plan to have a child if u + v ≥ ū, and will decide
not to have children otherwise (although unwanted pregnancies are always
possible).
The surplus generated by marriage can therefore be expressed as the
15
difference between this marital utility and the spouses’ welfare if single:
S (u, v) = max ((1 − p) y + p (z + u + v) , z + v + u)−max (y (1 − p) + p (z + u) , z + u)
(1)
In particular, when u + v < ū and u < ū, neither the couple nor the wife as
single want a child, and S (u, v) = pv. If u+v ≥ ū while u < ū, the couple has
a child while the wife as single would not, and S (u, v) = v − (1 − p) (ū − u).
Finally if u ≥ ū, both the couple and the wife, as single, would want a child,
and S (u, v) = v.16
Marriage Marriage cannot take place unless the surplus generated is non
negative. A first property is that the sign of the surplus depends only on the
man’s taste parameter; specifically, S (u, v) ≥ 0 if and only if v ≥ 0, as can
readily be seen from equation (1). This reflects the fundamental difference
between men and women in our model (and, arguably, to a large extent in
real life), namely that, as stated earlier, women do not need to be married
to become a mother, while men cannot enjoy paternity without being in a
couple. Therefore, for any man who valuates children positively, marriage is
Pareto improving, because a birth (wanted or not) would increase his utility
16
We do not consider the case u + v < ū while u ≥ ū. Indeed, this requires v < 0; but
then S (u, v) < 0 and marriage does not take place (see below).
16
by v at no cost for the wife; he would actually be willing to pay for this opportunity of accessing paternity. On the contrary, men who dislike children
(v < 0) will tend, everything equal, to remain single (although they may
sometimes be bribed into marriage), because singlehood is a simple way of
guaranteeing the absence of a child. Women, on the other hand, cannot lose
from marriage, even if they dislike children: in our model, the probability
of unwanted pregnancies is the same for married and single women. They
are at worst indifferent; moreover, they may in some cases strictly gain from
marriage, because of the compensation they may receive from their husband.
Therefore, any woman is a potential wife, while not all men are potential
husbands.
This basic difference has crucial consequences for the marriage market.
Indeed, the supply of potential wives is equal to the total female population,
irrespective of the distribution of the taste parameter. On the contrary, in
most cases only male with a positive preference for children will consider
getting married.17 This creates a systematic asymmetry on the marriage
market, whereby women tend to be in excess supply even when population
sizes are similar.
17
Note, however, that in some cases men with a ’low enough’ distaste for children can
choose to marry because they are compensated by the wife. This situation is discussed in
Subsection 4.2.
17
3.2
Stable matches
Characterization In our transferable utility world, a match is stable if
and only if it maximizes total surplus; therefore the existence of a stable
match is guaranteed by standard results (see for instance Chiappori and al
2007). Moreover, a crucial property of the surplus function is the following:
Proposition 1 The surplus function is supermodular: if u0 > u, v 0 > v,
then
¡
¢
¢
¡
¢
¡
S u0 , v0 + S (u, v) ≥ S u, v0 + S u0 , v
(2)
Proof. See Appendix
A well known consequence is:
Corollary 2 There exists a stable match which is assortative on taste for
children.
Since the surplus function is not strictly supermodular (i.e., one can find
u0 > u, v 0 > v such that (2) is satisfied as an equality), the stable match is
not unique in general. Indeed, if a stable match is such that u marries v, u0
marries v 0 and S (u0 , v 0 ) + S (u, v) = S (u, v 0 ) + S (u0 , v), then switching the
couples to (u, v0 ) and (u0 , v) maintains stability.
18
Our main interest is in how the spouses share the surplus generated by
marriage, and how this allocation is affected by an exogenous change in
birth prevention technology. As always with matching models, the qualitative properties of the stable match and of the resulting allocation of surplus
crucially depends on which side is in excess supply, although the notion of
’excess supply’ has a specific meaning in our context: as discussed above,
one should compare the total female population with the mass of men with
a positive preference for children. Formally, the marriage market is characterized by an excess supply of women if:
Z
0
V
g (t) dt = M − G (0) < 1
As argued before, if the male and female populations are of similar
sizes, and if a non negligible fraction of the male population dislike children, women will be in excess supply, a case we consider as the benchmark
situation in our analysis.
We may further characterize the stable matches in this benchmark case.
First, in any stable match, all men with a positive taste for children v are
married, and all men with a negative v are single. Also, we may define uM
19
by:
Z
U
uM
f (s) ds = M − G (0) =
Z
V
g (t) dt
0
In words, uM is such that the number of women with a taste parameter
larger than uM is equal to the number of men with a positive taste for
children.
Then two cases can be distinguished.
Proposition 3 (‘Traditional fertility’) Assume that uM < ū = y − z.
Then:
1. There exists a Strictly Assortative Stable Match (SASM) such that (i)
a woman with taste u is married (necessarily with a man with a non
negative taste parameter) if and only if u ≥ uM , (ii) if (u, v) and
(u0 , v 0 ) are two married couples and u0 ≥ u then v 0 ≥ v (strictly assortative matching), (iii) there exists a pivotal married couple (uP , vP ),
with uP > uM and uP + vP = ū, such that a married couple (u, v)
elects to have a child if and only if u ≥ uP .
2. For any stable match, no single woman wants children, and some married couples do not want children. Moreover, if a couple (u, v) is married and elects to have children in the SASM, then for any stable match
both u and v are married (although not necessarily together) and have
20
children.
Proof. See Appendix
In words, the condition uM < ū = y − z implies that the ‘last married’
wife uM is a conditional mother, who would not elect to have children unless compensated by her husband. In the SASM, only women above the
threshold uM get married. Moreover, uM women marry men unwilling to
compensate them for having a child, and these couples do not want children. The pivotal couple (uP , vP ) is indifferent to the presence of a child;
any couple whose taste for children is higher than the pivotal chooses to
have a child. Finally, single women are conditional mothers, therefore they
elect not to have a child; out-of-wedlock fertility is exclusively involuntary.
The picture is therefore one of traditional fertility, in which no single woman
(and not all couples) are voluntarily fecund.
The SASM is not the only stable match, because S is not always strictly
supermodular. Specifically, take two couples (u, v) and (u0 , v 0 ) such that
u0 > u, v0 > v. If both couples want children (i.e., u+v ≥ ū), then S (u0 , v 0 )+
S (u, v) = S (u, v 0 ) + S (u0 , v), and there exists a stable match in which u
marries v 0 and v marries u0 . The same holds true if u0 + v0 < ū (that is, both
couples decide not to have children). Therefore, among couples who decide
to have children, the identity of the spouses is indeterminate, and the same is
21
true among couples who do not want children. Moreover, women who marry
but do not want children are indifferent with remaining single; therefore the
set of married women also varies across stable matches. However, the set of
women who want children once married is constant across stable matches.
The alternative situation is described in the next result:
Proposition 4 (‘Father shortage’) Assume, conversely, that uM > ū.
Then for any stable match, all married couples want a child; moreover,
some single women also decide to have a child. Also, there exists a Strictly
Assortative Stable Match such that (i) women with a taste parameter equal
to uM marry men with a taste parameter equal to 0, (ii) all women with a
taste parameter larger than uM are married with men with a taste parameter
larger than 0, and (iii) all women with a taste parameter smaller than uM
are single.
Proof. See Appendix
In this alternative, ‘father shortage’ situation, the number of men willing to marry and have a child is so small that the last married woman is
an unconditional mother, who wants a child even without any compensation from her husband. Then all married couples want children. Moreover,
some unconditional mothers remain single, and choose to have a child as
well, resulting in voluntary out-of-wetlock births. Again, the identity of the
22
married women is not determined at equilibrium, because all unconditional
mothers are equivalent from a male’s perspective.
Finally, it is crucial to note that the utility level of any individual is the
same in all stable matches. Since our main interest is in assessing changes
in individual welfare, we can, without loss of generality, concentrate on one
particular stable match for our analysis. It is natural to choose the SASM,
which we do throughout the paper. For any married couple (u, v), the mass
of men with a taste parameter v 0 > v is then equal to the mass of women
with a taste parameter u0 > u :
1 − F (u) = M − G (v)
It follows that the mapping ψ which associates to any married man v the
taste parameter u of his spouse is given by u = ψ (v) = F −1 (1 − M + G (v)).
Equivalently, we can define φ = ψ −1 by v = φ (u) = G−1 (M − 1 + F (u)).
Both φ and ψ are increasing. In particular, for any stable match, the total
surplus generated by children, u + v, is increasing in u.
Surplus distribution In our continuous framework, the allocation of surplus between household members is exactly pinned down by the stability
condition: competition on the marriage market imposes a specific distribu-
23
tion of welfare between spouses. This allocation is described in the following
Proposition. Not surprisingly, the two cases described in propositions 3 and
4 lead to qualitatively different allocations:
Proposition 5
1. In the ‘father shortage’ situation uM > ū, all the
surplus is received by the husband; each married woman has the same
utility as when single.
2. In the alternative, ‘traditional fertility’ situation uM ≤ ū: in subpivotal couples (u + v < ū), who do not want children, all the surplus
S = pv goes to the husband, and the wife is indifferent between marrying and remaining single; in super-pivotal couples (u + v ≥ ū), who
have children, the husband receives SM = v − (1 − p) (ū − uP ), and the
wife receives SW = (1 − p) (min (u, ū) − uP ) .
Proof. See Appendix
The intuition for these results is clear. Consider father shortage. The
last married woman is an unconditional mother, who needs (and receives)
no compensation for deciding to have a child. Competition between women
requires that all husbands be in the same situation, namely having a child
at no cost, thus pocketing all the surplus generated by fertility.
24
The alternative, ‘traditional fertility’ case is slightly more complex. Consider first the ‘pivotal’ couples, for which u + v = ū. Pivotal women need
to be compensated to have a child; specifically, the compensation, in that
case, equals both the minimum the wife would require and the maximum
the husband is willing to pay to have a child. Couples with preferences for
children smaller than pivotal choose not to have a child; the wife is not
compensated, and the husband enjoys the benefits of involuntary births for
free. In super-pivotal couples, on the other hand, the wife’s compensation
is pinned down by competition, giving the patterns described. In practice,
all super-pivotal husbands transfer a fixed amount to their wife, equal to the
compensation required by the pivotal woman, namely (1 − p) (ū − uP ).
Each spouse’s total utility follows. The ’father shortage’ case is straightforward: each husband consumes his income Y and receives the benefit v
derived from the child, while the wife’s utility is as if she was single. In
the alternative case, in sub-pivotal couples (u + v < ū), who do not want
children, the husband’s utility is Y + pv, and the wife receives her utility as
single, namely y(1 − p) + p (z + u); in super-pivotal couples (u + v ≥ ū), who
all have children, the husband’s utility is Y + v − (1 − p) (ū − uP ), and the
wife’s is UW = z + u + (1 − p) (ū − uP ) .18
18
If men with a positive utility from children outnumber the female population, then
all women are married and stability requires that each married man be indifferent with
25
3.3
Comparative statics
Simple as it may be, our model still offers interesting insights on the role
of several key parameters. We simply describe the main conclusions; the
reader is referred to Chiappori and Oreffice (2006) for a detailed discussion,
including the proofs.
Incomes Consider, first, the impact of income changes on fertility and allocations. In our quasi linear world, changes in male income has little impact
beyond increasing men’s utility. On the other hand, female utility always
increases with both y (her income without a child) and z (her income with
a child). Assume, now, that y and z are inflated by the same factor, so that
the difference ū = y − z increases proportionally; one may think of a general
increase in female income. Then the ’pivotal’ parameter uP is increased, although by a smaller amount.19 The number of super-pivotal couples is thus
reduced. Within such couples, the transfer received by the wife is increased;
intuitively, the compensation has to be raised because of the larger opportunity cost incurred when having a child; the precise amount depends on
the distribution of taste parameters in the population. We conclude that (i)
remaining single. Therefore, the utility men derive from having children is captured by
women under the form of additional consumption
19
1
P
Indeed, 0 < ∂u
∂ ū = 1+φ0 (uP ) < 1.
26
total fertility is reduced, (ii) in non fertile couples, the wife’s utility grows
by exactly her gain in income, and (iii) in voluntarily fertile couples, her
gain is larger than the additional income, since the compensation paid by
the husband increases as well; the raise in income is thus compounded by
intrahousehold effects.
Single parent benefits So far, we have assumed that the income of a
woman with a child is z whatever her marital status. We now allow for
single parent benefits, say b, and assume for simplicity that all single women
are eligible. The income of a woman with a child is thus still z when she is
married, but becomes b + z when single. This fact has various consequences.
First, out of wedlock fertility generally increases; indeed, the parameter
threshold for single motherhood drops to ū − b. Fertility decisions within
marriage are not affected. Secondly, marriage is discouraged; specifically, the
pool of potential husband is decreased because men will not marry unless
their taste parameter v is at least b (so that, from a collective viewpoint, the
gain in husband’s welfare more than compensates the loss in income resulting
from marriage). Therefore, the ‘father shortage’ situation is more likely to
occur than before. Then married women are exactly compensated for their
income loss - i.e., they receive a transfer equal to b, and the husband’s surplus
27
is v − b. Finally, in the alternative, ‘traditional fertility’ case, sub-pivotal
women receive a transfer equal to pb, compensating them for the expected
loss in income. Super-pivotal women have a child and receive a transfer
equal to (1 − p) (ū − uP ) + pb.
In summary, the introduction of single parent benefits will in general
profit all married women. There is however an exception to this statement.
If the resulting equilibrium switches from traditional fertility to father shortage, the increase in married women’s reservation utility due to single parent
benefits is partly or totally offset by the loss of the surplus they previously
received. Then the impact on female welfare is ambiguous and depends on
the particular distributions.
Smaller male population Let us now study the impact of a reduction
in the number of males with a positive taste for children. Assume that
the distribution G is replaced with some G0 such that G0 (U ) − G0 (t) ≤
G (U ) − G (t) for all t. The mapping φ is then replaced with some φ0 such
that φ0 (u) < φ (u) for all u; i.e., each woman is matched with a man with
lower taste for children. The marriage threshold uM is increased; in particular, a situation of father shortage is more likely to occur. If the equilibrium switches from traditional fertility to father shortage, total fertility
28
decreases, but (voluntary and involuntary) out-of wedlock fertility increases
significantly (some single women now want children); all married women
suffer since their share of the surplus is reduced to zero, while married men
all gain. If, on the contrary, the equilibrium remains within the ‘traditional
fertility’ class, women’s share of surplus is still reduced. Namely, the new
pivotal couple (u0P , vP0 = φ (u0P )) is such that u0P > uP and vP0 < vP ; the
number of sub-pivotal women, who receive no surplus, is increased. Within
super-pivotal couples, the transfer to the wife is reduced to (1 − p) (ū − u0P ).
Total fertility decreases (less couples want to have children), but (involuntary) out-of-wedlock fertility increases.
In practice, these predictions seem to fit fairly well the main stylized facts
characterizing the marriage market in the US. The raise in female income
over the last decades has coincided with a reduction in fertility. Moreover,
the idea of an amplification of these gains through intrahousehold transfers
may explain the stronger demand for college education among women, as
argued by Chiappori, Iyigun and Weiss (2006). Also, an overwhelming phenomenon that took place during the last decades is the dramatic raise of
incarceration rates for young, low-skilled black males.20 If we accept that
20
According to Western and Pettit (2000), the percentage of young, high-school dropouts
black males in prison or jail reached 36% in 1996, up from 15% in 1982; as a comparison,
the rate for young, high-school dropouts white males went only from 4% to 7% over the
same period. Similar effects can be observed on employment data. Among black high-
29
the marriage market is largely assortative by race and education, the ’submarket’ of black, low-skilled individuals has experienced a dramatic decline
in male supply. According to our analysis, such a brutal change should induce a drop in marriage but a sharp rise in non marital fertility. Indeed,
while the overall fertility rate of black, high-school dropout women aged 25
to 29 has slightly declined over the period (from 102.5 per mil in 1980 to
99.5 per mil in 200021 ), the fraction of women never married with children
has raised from 3% in 1960 to 35% in 1990, as compared to 6% for both
white high-school dropouts and black women with college education in the
same age range.22 Our results, on this point, are in line with earlier intuitions proposed by Neal (2004). Also, an implication of our model is that
the welfare of unskilled black women may have significantly decreased over
the period.
school dropouts aged 26-36, the percentage of individuals who worked at least 26 weeks
during the poll year dropped from 80% in 1960 to 44% in 1990. For high school graduates,
the drop is from 86% to 67% (Neal, 2004, table 2). See also Wilson (1987).
21
Computation made by the authors using National Vital Statistics Report, 50-5, 2002,
Table 4, p.30; Monthly Vital Statistics Report, 31-8, 1982, Table 19, p. 30; National
Vital Statistics Report, 50-5, 2002, Table 21, p. 51; “Educational Attainment, US Census
Bureau, Historical Tables, Table A-2, p. A-4.
22
The effect may initially have been further boosted by the increased generosity of single
parent benefits over the 60s and the early 70s. Note, however, that this latter trend has
been largely reverted since then (Moffit 1992), while the increase in birth outside marriage
has persisted, suggesting that the marriage market factor may have been dominant.
30
4
Changes in the birth control technology
We now come to the main implications of our model, namely the impact of
a technological change in birth control.
4.1
Legalizing abortion
We first consider the impact of innovations in birth control technology that
reduce the probability of unwanted pregnancies, assuming that all women
(including single) are given free access to the technology; a natural example
could be the legalization of abortion that took place in the 70s. In our
model, we consider the impact of a reduction in the probability of unwanted
pregnancies from p to some p0 < p.
The model generates the following conclusions:
• Not surprisingly, single women who do not want a child benefit from
the technology, precisely because unwanted pregnancies become less
likely. The gain for a single woman of taste u < ū is thus (p − p0 ) (ū − u).
• Regarding married women, the outcome depends on the type of equilibria. In the extreme case of father shortage, the distribution of the
surplus generated by marriage remains unbalanced in favor of the husband; each wife receives her utility as single. Since married women are
31
all unconditional mothers that would not use the birth control technology, the legalization of abortion has no impact on married couples.
• The alternative, traditional fertility case is much more interesting.
Consider, first, subpivotal couples. Because better birth control is
now available, the total surplus generated by marriage is reduced.
The wife still receives her utility as single, which has however been
increased by (p − p0 ) ū by the new technology. Meanwhile, the father’s
expected utility is reduced by the same amount, since the windfall gain
from unwanted pregnancies has become less likely. We conclude that
subpivotal couples experience a decline in fertility; moreover, the wife
gains, and the husband loses, from the new technology.
• Consider, now, superpivotal couples. Their fertility decision is not affected by the new technology; these couples want children and will not
resort to abortion. This, however, does not mean that the reform has
no impact on these couples, because the intrahousehold allocation of
the surplus is significantly altered. Husband-to-wife transfers increase
by (p − p0 ) (ū − uP ); since behavior is unaffected, her utility is boosted
(and his declines) by the same amount.
The intuition for the last result is that the intrahousehold distribution
32
of resources is driven by the marginal woman. Except for the extreme case
of father shortage, she is indifferent between getting married and remaining
single without (wanted) children. Her reservation utility is thus improved
by the new technology. The nature of a matching game implies that any
improvement of the marginal agent’s situation must be transmitted to all
agents above the marginal one. All in all, the new technology reduces fertility of some couples (then the wife grabs all the benefits); moreover, in
those couples whose fertility does not change (and who do not use the new
technology), the reform results in a net transfer from the husband to the
wife. In particular, it is the case that all men lose from the new technology.
We thus conclude that in our model an improvement in the birth control
technology, such as the legalization of abortion, generally increases the welfare of all women, including those who want a child and are not interested
in the new technology. Note, however, that the mechanism generating this
gain is largely indirect. The reason why even married women willing to have
a child benefit from the birth control technology is that the latter, by raising
the reservation utility of single women, raises the ’price’ of all women on the
matching market. Moreover, this logic fails to apply in situations of severe
shortage of marriageable men. As argued above, empirical evidence suggests
that specific submarkets (e.g., low skill minority women) may exhibit the
33
typical features of a father shortage. Our analysis suggests that married
women belonging to such social groups may have derived little benefits from
the legalization of abortion.
4.2
The power shift of the pill
The previous argument shows clearly that an important channel through
which the new technology benefits all women - what could be called ‘female empowerment’ - is the raise in their reservation utility. In turn, the
source of this change lies in the fact that single women can access the new
technology. Consider, now, a situation in which a new technology of this
type is introduced but exclusively available to married women. A typical
illustration is provided by the pill, which became available first (in 1960) to
married women only, and later (by the end of the 1960s) to single women as
well (Goldin and Katz 2002).
We therefore consider a situation in which only married women can access a technology that reduces the probability of unwanted pregnancies from
p to some p0 < p. The analysis is qualitatively different from the previous
case. We do not provide here an exhaustive characterization; the reader is
referred to Chiappori and Oreffice (2007) for precise statements and proofs.
Let us simply indicate the main qualitative patterns of the stable match in
34
this context. First, the surplus function is still (weakly) supermodular. As a
consequence, matching is still positive assortative: among married couples,
women with a larger taste for children marry men with a larger taste for
children. Secondly, the set of potential husbands is increased. Indeed, the
surplus S (u, v) can now be positive for negative taste parameters v, provided that the wife’s parameter u is below ū. The intuition is that women
who do not care for children are now willing to ’purchase’ the technology
through marriage, and are willing to pay the husband for the reduction in
unwanted pregnancies. This compensation attracts new possible mates. Finally, the set of married women qualitatively differs from the previous case;
it now consists (in general) of two disjoints intervals. As before, women with
a high preference for children marry and have kids. At the other end of the
spectrum, women with a minimum taste (or maximum distaste) for children
also marry, precisely because they want access to the technology. In between
is an interval of women with ’intermediate’ taste parameters; they are the
ones who remain single. Without the innovation, some of them would have
married and had children: the innovation thus reduces both wanted and
unwanted fertility.
In terms of welfare, women who do not want children and marry to
access the new technology gain from the innovation, although part of the
35
gain is repaid to their husband. Some women are single with or without the
new technology, and are therefore indifferent. All other women strictly lose
from the innovation. Some are now single, while they used to be married
and receive a fraction of the surplus generated by marriage. Others would
marry irrespective of the innovation; however, their share of the surplus is
decreased by the innovation. In particular, one can show that the benefits of
a larger pool of potential husbands cannot offset the loss due to the increased
competition from low taste women that generates it. Finally, all men benefit
from the innovation.
In other words, making the new technology available to married women
only has the major, and somewhat counter intuitive consequence of reversing the empowerment effect, for two reasons. First, the previous conclusions
that all women benefit from the new technology rely on a crucial insight of
matching models - namely, that the welfare of the marginal (’last single’)
woman drives the equilibrium allocation. If the new technology is exclusively reserved to married women, this effect disappears, and with it the
redistribution generated by the innovation. In addition, the availability for
marriage of women with a low taste for children, who are willing to compensate their husband for getting married and gaining access to the new
technology, toughens competition for husbands. Therefore, women with a
36
higher taste for children lose from the introduction of the new technology.
Only women with a very low taste parameter gain from the innovation.
The discussion above emphasizes the complex and partly paradoxical
welfare impact of a new technology. On the one hand, its effects can go well
beyond the individuals who actually use it, or even consider using it. Our
model suggests that a major effect of legalizing abortion may have been a
shift in the intrahousehold balance of powers and in the resulting allocation
of resources, even (and perhaps especially) in couples who were not considering abortion as an option. On the other hand, the new technology benefits
all married women only because it is available to singles. A technological
improvement which is reserved to married women will have an impact on
their fertility, partly because it changes the mechanisms governing selection
into marriage. But its impact on women’s welfare is either nil or negative,
except for a small fraction of women who choose marriage as an access to
the new technology.
Obviously, this analysis is only partial, in that it omits other benefits
of the pill (such as women’s increased ability to plan their fertility and to
achieve higher levels of education). These aspects, which have been intensively discussed in the literature, clearly favored all women, including
married one. Still, the strong message of this analysis is that reserving the
37
technology to married women makes an important difference, and, more
surprisingly, that most married women are likely to lose from this exclusivity. Our analysis thus suggests that as far as the intrahousehold balance of
power is concerned, the introduction of the pill in 1960 (when it was available to married women only) may have had a disempowerment effect on
most women. The true empowerment revolution came later; it was caused
by the generalization of its availability for single women during the late 60s,
and strengthened by the legalization of abortion in the 70s. In particular,
the true empowerment effect of Roe vs. Wade may have come less from the
innovation itself than from its availability to all women, including singles.
4.3
Some extensions
We briefly discuss possible extensions of our results.
Costly access to the new technology As a first extension, consider the
case in which the new technology, while available to all women, is ‘costly’.
The cost, here, should be understood in a general way; it includes financial
costs, but also the moral or ethical discomfort some women may experience
with the new technology. We assume for the moment that the cost is identical for all women. Since, however, preferences for children differ, a uniform
cost generates different responses over the population. Specifically, single
38
women will use the technology only if the net cost of the child (i.e. the income loss minus the benefit u) is large enough to compensate for the abortion
cost. Therefore, some single women who do not want children will nevertheless decline to use the new technology; only those with a small enough
preference for children will. Similarly, among subpivotal couples, some (and
possibly all) will be unwilling to pay the cost. The resulting equilibrium
is therefore in-between the initial, pre-legalization benchmark and the free
technology case. In particular, in terms of welfare, all women are weakly
better off than before the technology became available, although some of
them may actually be indifferent; but all women are either indifferent or
worse off than if the technology was available at zero cost
A clear policy implication is that any legislative change that increases the
cost of the technology reduces in general welfare of all women, including those
who would not use (or even consider using) the technology. For instance,
in the first few years after the national legalization in 1973, abortion was
eligible for Medicaid public funding; this provision was ruled out in 1976 by
the Hyde Amendment that generated many similar provisions at the state
level. According to our analysis, these fluctuations in public funding23 not
only modify women’s actual use of abortion, but affect the gains generated
23
Other examples include the provisions of mandatory counseling and of parental consent for abortion on minors.
39
by the legalization for all women - including those who are willing to bear
the costs and those who are not interested in abortion in any case.
Heterogenous costs In practice, different women face different costs.
The psychological distress clearly varies between women; and even though
the financial cost is more uniform, its relative impact on individual consumption and welfare is as heterogenous as individual incomes, a feature that our
simplified model does not consider. To capture the idea of heterogenous
costs, one can make the simple assumption that some women pay the full
cost c discussed above, while for others the cost is nil. Also, we assume that
the distribution of this cost among women is not correlated with marital
status.24 To further sharpen the discussion, consider the extreme case of
an infinite cost. Hence, among women with identical preferences some are
willing to adopt the new technology while the others do not (or cannot);
and the respective proportion of the two classes is independent of the taste
parameter u.
In this new context, women who do not accept the new technology are in
a weaker position on the market, because their reservation utility as single is
lower. A first consequence is that these women are, everything equal, more
24
Under the opposite extreme assumption where abortion is costly for single women
only, we are back to the case, studied above, of a technology exclusively available to
married women.
40
likely to marry. Regarding the equilibrium structure, different cases should
be considered. In a ’father shortage’ situation, first, nothing is changed,
since the marginal married woman wants children anyway. In the alternative, ’balanced fertility’ case, things are more complex. Clearly, women who
would not use the technology are more attractive to men who value children,
because of the possibility of involuntary pregnancies. This fact may have
different consequences depending on the parameters. For instance, there
may exist two marginal married women, one who accepts the technology
while the other does not; the taste parameter is larger for the former. Alternatively, it may be the case that all women who reject the technology
are married, while some abortion-prone women remain single. From a welfare point of view, it should however be noted that no woman can possibly
lose from the introduction of the new technology, even when some (possibly
many) women reject its use. It is in general the case that all women gain
from the introduction, including those who reject it (the exception being the
’father shortage’ case). Conversely, if one takes as a benchmark the situation
in which all women can access the technology, the introduction of costs that
preclude its use by some women harms all women in general.
41
4.4
Shotgun marriages
In the discussion of the consequences of Roe vs. Wade, the issue of ‘shotgun
marriages’ has attracted considerable attention. In an influential paper, Akerlof, Yellen and Katz (1996) have argued that abortion led to the disappearance of shotgun marriages, since women could avoid unwanted pregnancies,
a fact that was known to (and potentially used by) men. They conclude
that legalizing abortion may actually have harmed some women.
‘There is no such thing as a free marriage’ A first difference between
AYK’s model and ours reflects the emphasis we put on the intrahousehold
allocation of resources as the endogenous outcome of equilibrium formation
on the marriage market. In our model, a women being ‘shotgun married’ is
not necessarily better off than a woman remaining single, especially if the
latter can use the new technology. The intuition is simply that if ’marriageable’ men are in short supply, the surplus generated by marriage will be
partly or fully appropriated by the husband; the woman will thus ‘pay’ for
marriage by a low share of household consumption. While this extreme conclusion is clearly linked to our simple setting (e.g., the absence of frictions,
hence of post-marital bargaining), we still believe that it stresses an important issue - namely, that shotgun marriage may not come for free. Whether
42
forced marriages closely following an unwanted pregnancy really benefited
women is at least debatable; after all, the resulting allocation of household
resources was unlikely to favor women, and it is at least conceivable that
the new wife’s situation within such ‘shotgun couples’ was no better than
what it would have been had she been single.25
Still, an important insight of AYK can readily be incorporated into our
discussion. Specifically, assume, following AYK, that in the absence of the
birth control technology, social norms impose an implicit commitment from
the male part, whereby sexual activity leading to pregnancy must end up in
marriage, even against the male’s initial intention. Assume, moreover, that
the availability of abortion results in the disappearance of the social norm, in
the sense that the father of an unwanted child no longer feels committed by
the mother’s decision to keep the child. Under such circumstances, the new
25
The issue is difficult to assess empirically, if only because modifications of social norms
are hard to document (let alone measure). Note, however, that a complete investigation
must involve estimates of intrahousehold inequality. An analysis that neglects the changes
in intrahousehold allocations resulting from the new technology misses a key issue, and
may therefore lead to erroneous conclusions. To take but one example, the idea of a
‘feminization of poverty’ taking place in the 70s should probably be considered with some
caution. Insofar as available empirical evidence mostly relies on comparison between individuals and couples while failing to address the crucial issue of the allocation of individual
well-being within couples, it may be largely misleading. Standard answers to this problem,
based on equivalence scales, are inadequate. Relating the income of a single mother to
half (or, for that matter, any fixed fraction) of the couple’s income amounts to assuming
that income is split within the couple according to some exogenously given rule. Our
paper points precisely to the opposite direction: economic theory in general, and equilibrium considerations in particular, tells us that the split is endogenous, driven by the
environment, and responsive to the technological changes as well as to market conditions
in general. Should these effects be taken into account, the conclusions may be reversed.
43
technology may change the distribution of taste for children in the male
population. Specifically, men may accept (involuntary) fatherhood as an
indissoluble aspect of sexual activity. Once the link has been broken, male
preference for children can be expected to decline, at least for a fraction
of the male population. As discussed above, such a decline will typically
harm women, and may partly (or even totally) offset the benefits of the new
technology. Additionally, the reduction in shotgun marriage could provide
an explanation for the increase in out-of-wedlock fertility, since pregnancies
no longer result in marriage once abortion has been legalized.
The distinction between our model and the variant just sketched involves
deep social and ethical issues. In a sense, one approach views children
mostly as the intended consequence of a well-informed decision (which can
be thought of as investment in human capital), whereas the other emphasizes
fertility as an involuntary by-product of sex. Still, empirical considerations
may also be invoked in this debate. One of the best empirical discussion
of the ‘shotgun’ analysis is provided by Neal (2004). Neal makes two main
points. First, we expect social norms to influence most or all members of
a given society. If one believes that the decline in shotgun marriages is
the primary explanation for the rise in never-married motherhood, one has
to explain why its impact in terms of out-of-wedlock births appears to be
44
concentrated among a very specific fraction of the population, principally
less educated black women. Secondly, if most of these additional births are
unwanted, as implied in the AYK version, one would expect a sharp increase
in the number of adoptions per non marital birth over this period: even
if many women are unwilling to terminate unwanted pregnancies through
abortions, whether for ethical, religious or financial reasons, relinquishing
the child to adoptive parents remains an option. Neal shows, however, that
this prediction is counterfactual: the adoption rate has, if anything, declined
over the period.26
Finally, the ‘impoverishment of women’ effect described by AYK must
operate through a particular channel; namely, a significant fraction of the
male population decides to remain single, while they would have chosen (or
be forced) to get married before legalization. One way to empirically analyze
this issue, hence, is to see whether legalization has significantly increased the
probability of singlehood in the male population. Casual observation does
not support this prediction. The proportion of single men in the male population does not seem to respond to legalization.27 Actually, when regressing
26
Note that our model provides simple answers for both phenomena. We relate the raise
in out-of-wedlock fertility to the sharp decline in men’s supply, a phenomenon specific to
the population of black males with lower education. Moreover, in our model the babies
born out-of-wedlock were wanted, which explains the feeble rate of adoptions.
27
Whites or blacks, various age ranges, evidence from the annual March Census Current
Population Reports ”Marital status and living arrangements”, 1968 to 1979. See Chiappori
and Oreffice (2006) for detailed figures.
45
a male singlehood dummy on age, education, race and fixed effect by year
and state, as well as an abortion dummy,28 on the male population aged
15-50 in the CPS March supplements 1968-1980, we find that the abortion
dummy is not significant, and its point estimate is actually negative.
5
Final comments
The model proposed in this paper provides a simplified view of the phenomena at stake. Many issues remain open. Marriage markets are characterized
by multidimensional heterogeneity (tastes, but also incomes, ...). Frictions
are paramount, which implies that intrahousehold bargaining could profitably be taken into account.29 Dynamic issues are crucial, not only because divorce and remarriage are important features of the market, but also
because such factors as the average age at marriage or the age difference
between the spouses are known to matter for equilibrium. A single, unified
market for marriage probably does not exist; the relevant concept is more
a multiplicity of markets by age, localization, race and religion. Our model
applies to each of these submarkets, with possibly different conclusions in
28
The abortion dummy variable takes a value of unity in 1970 and on for observations
located in any of the five states that fully legalized abortion in that year (California, New
York, Washington, Alaska, Hawaii) and for all observations after January 1973 following
the Supreme Court’s ruling on Roe vs. Wade; it is zero otherwise.
29
This view is explored by Chiappori and Weiss (2003, 2005) and Chiappori, Iyigun and
Weiss (2005).
46
the various situations. For instance, the decrease in the supply of men may
be a more stringent phenomenon in some contexts (say, younger population in impoverished urban neighborhoods) than in others - and this fact
may explain considerable differences in marriage or out-of-wedlock fertility
across the population. The empirical definition of these submarkets and
the measure of their interconnections will raise delicate problems for future
research.
Despite these simplifications, we believe that our approach puts forth
important insights regarding the impact of birth control technologies. Our
main message is that issues related to intrahousehold allocation are of crucial importance, and that the distribution of resources and welfare within
couples must be understood as an endogenous phenomenon which responds
to changes in the economic and technological environment. An important
consequence is that the introduction of new technologies (or new benefits,
for that matter) may have a major impact on individual welfare even within
couples who do not directly benefit from it. This intuition has been frequently
mentioned by sociologists (the idea of ‘female empowerment’ resulting from
the legalization is an old theme of feminist studies), but it may have been
somehow disregarded by economists, at least as far as explicit modeling is
47
concerned.30
Economic analysis provides new and interesting insights on the ‘female
empowerment’ issue. For instance, the impact of a given reform deeply varies
with such exterior determinants as female wages, the generosity of benefit
systems or the situation on the market for marriage. These interactions are
complex. For instance, our model suggests that there is no simple relationship between the extent to which an innovation is used and the magnitude
of its impact on intrahousehold inequality.31 Studies analyzing the fertility consequences of the innovation, although important, are no substitutes
for an investigation of intrahousehold allocation issues. This is a task for
which specific models must be devised. A final contribution of our model is
precisely to show how the ‘collective’ analysis, which has been largely developed at the household level, can easily be incorporated into a market-wide
analysis to provide a model of the type required. In a sense, the link is very
natural. The ‘sharing rule’, a crucial ingredient of the collective approach,
30
An interesting and early exception is the analysis of guaranteed employment programs
in India proposed by Haddad and Kanbur (1990). See also Grossbard-Schechtman (1993)
and Edlund and Korn (2002).
31
Assume a given innovation (say, abortion legalization) is simultaneously introduced
in two different ‘submarkets’, one experiencing a father shortage while the other is in a
situation of balanced fertility (in the sense defined in the paper). The consequences on
fertility will be much larger in the first submarket (all ’conditional mothers’, being single,
will use the technology), whereas the consequences on intrahousehold allocation is nil.
In the second submarket, fewer women use the technology, but the allocative impact is
maximum.
48
has a direct translation in the context of a matching model; specifically,
it defines the payoffs that stabilize a given match. Because our approach
is based on matching, it seems well adapted to analyze the impact of the
reform on couples formed after the reform has been implemented; it thus
complements alternative works based on bargaining models.32 Our contribution should be seen in the perspective of a general line of research which
is currently pursued.
32
See for instance Chiappori and Donni (2005)
49
APPENDIX
A
Proof of Proposition 1
Define
¡
¢
¡ ¡
¢
¡
¢¢
D = S u0 , v0 + S (u, v) − S u, v0 + S u0 , v ≥ 0
where u0 > u and v 0 > v. Equivalently,
D = Γ(u0, v0) + Γ(u, v) − (Γ(u, v0) + Γ(u0, v)) ≥ 0
where Γ(u, v) represents the aggregate utility of the couple (u, v).
We know that if (u, v) have children so do (u0 , v) , (u, v 0 ) and (u0 , v 0 ); and
that if (u0 , v 0 ) do not have children neither do (u0 , v) , (u, v 0 ) and (u, v). We
therefore consider several cases.
1. All couples have children, or none of the couples have children: then
D=0
50
2. (u0 , v 0 ) have children, but none of the others; then
¢
¡
¢
¡
Γ u0 , v 0 = z + Y + v0 + u0 ≥ y + Y + p v 0 + u0 + z − y
¡
¢
¢
¡
Γ u, v 0 = y + Y + p v0 + u + z − y ≥ z + Y + v0 + u
¢
¡
¢
¡
Γ u0 , v = y + Y + p v + u0 + z − y ≥ z + Y + v + u0
Γ (u, v) = y + Y + p (v + u + z − y) ≥ z + Y + v + u
hence
¢¢
¡
¡
D = − (1 − p) y − z + v 0 + u0 ≥ 0
since u0 + v 0 ≥ y − z = ū by assumption.
3. (u0 , v 0 ) and (u, v 0 ) have children, but none of the others
¡
¢
¡
¢
Γ u0 , v 0 = z + Y + v0 + u0 ≥ y + Y + p v 0 + u0 + z − y
¡
¢
¡
¢
Γ u, v 0 = z + Y + v0 + u ≥ y + Y + p v0 + u + z − y
¡
¢
¡
¢
Γ u0 , v = y + Y + p v + u0 + z − y ≥ z + Y + v + u0
Γ (u, v) = y + Y + p (v + u + z − y) ≥ z + Y + v + u
hence
¡
¢
D = (1 − p) u0 − u > 0
51
4. (u0 , v 0 ) and (u0 , v) have children, but none of the others: same argument as 3.
5. All have children except (u, v)
¡
¢
¢
¡
Γ u0 , v 0 = z + Y + v0 + u0 ≥ y + Y + p v 0 + u0 + z − y
¡
¢
¢
¡
Γ u, v 0 = z + Y + v0 + u ≥ y + Y + p v0 + u + z − y
¡
¢
¡
¢
Γ u0 , v = z + Y + v + u0 ≥ y + Y + p v + u0 + z − y
Γ (u, v) = y + Y + p (v + u + z − y) ≥ z + Y + v + u
D = (1 − p) (y − (v + u + z)) ≥ 0
since u + v ≤ y − z = ū by assumption.
B
Proof of Proposition 3
The existence of a strictly assortative match follows from supermodularity.
In such a match, all married women are above the threshold uM such that the
number of such women equals the number of ‘marriageable’ men (i.e., men
with a positive taste for children); in particular, uM women are married
with men with a zero taste for children, and since uM + 0 = uM < ū
these couples do not want children. Since the two distributions of male and
52
female are atomless, the sum u + v is continuous and strictly increasing over
married couples. Some couples will have children (since some mothers are
unconditional), while others (like (uM , 0)) will not, hence by continuity the
existence of a pivotal couple.
Consider, now, an arbitrary stable match. Assume that a single woman
wants a child, hence belongs to the unconditional mother category. Since
all men with positive taste are married, some must be married with conditional women, while they could receive a higher surplus by marrying the
unconditional single, a contradiction.
Finally, assume that a couple (u0 , v 0 ) is married and elects to have children in the SASM (so that u0 ≥ uP ), and there exists a stable match in
which u0 is single. In that match, some married man are married with a
wife u < uP and do not want children, while they could generate a higher
surplus marrying u0 , a contradiction. By the same token, v 0 must be married
for any stable match. Assume, alternatively, that there exists a stable match
in which u0 is married (say, to v) but does not want children; necessarily
v < ū − u0 ≤ v 0 . If v 0 is married to some u and does not want children, then
u < ū − v 0 ≤ u0 , and by condition 2 of the previous proof stability conditions
are violated. If v 0 is married to some u and elect to have children, then by
condition 3 of the previous proof stability conditions are violated.
53
C
Proof of Proposition 4
Again, the existence of a strictly assortative match follows from supermodularity. In such a match, all married women are above the threshold uM
such that the number of such women equals the number of ‘marriageable’
men (i.e., men with a positive taste for children); in particular, uM women,
who are unconditional mothers, are married with men with a zero taste for
children and want children. Consider, now, an arbitrary stable match. Some
unconditional mothers must be single. Assume that a married couple does
not want children; the husband could generate a higher surplus by marrying
an unconditional single mother, a contradiction.
D
Proof of Proposition 5
In a married couple (u0 , v0 ), where v 0 = φ (u0 ), let SW (u0 ) denote the part of
surplus going to the wife, and SM (v 0 ) = S (u0 , v 0 ) − SW (u0 ) the husband’s
share. Note, first, that stability requires that for any v
¡
¢
¡ ¢
S u0 , v ≤ SW u0 + SM (v)
54
with an equality for v = v 0 . Therefore
¡ ¢
¡ ¡
¢
¢
SW u0 = max S u0 , v − SM (v)
v
and by the envelope theorem:
¡ 0 ¢ ∂S (u0 , φ (u0 ))
0
u =
SW
∂u0
We start with the case of father shortage. For all married couples,
S (u, v) = v, hence ∂S/∂u = 0 and SW (u) is constant. Since it is zero
for the marginal wife uM , it is zero for all women.
Consider, now, the alternative case of balanced fertility. For subpivotal
couples (u < uP ), S (u, v) = pv, hence ∂S/∂u = 0. Again, SW (u) is constant, and zero for the marginal wife uM ; we conclude that SW (u) = 0 for
u ≤ uP . If u > uP but u ≤ ū, then
S (u, v) = v − (1 − p) (ū − u)
0 (u) = (1 − p). Since S (u ) = 0 we have S (u) =
hence ∂S/∂u = SW
W
P
W
(1 − p) (u − uP ). Finally, if u > ū then S (u, v) = v, hence ∂S/∂u = 0
and SW (u) is constant. Since SW (ū) = (1 − p) (ū − uP ) we have SW (u) =
55
(1 − p) (ū − uP ) for all u > ū.
56
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